Hello, Chief65!

You want to divide a bar of chocolate into $\displaystyle 1\times1$ squares.

The starting bar is an $\displaystyle m \times n$ rectangle.

Show that it can be divided by cutting a minimum of $\displaystyle mn-1$ times.

Each cut is a straight line parallel to one of the sides of the piece. Code:

*---*---* - - - *---*
| | | | |
*---*---* - - - *---*
| | | | |
*---*---* - - - *---*
: : : : :
: : : : : m rows
: : : : :
*---*---* - - - *---*
| | | | |
*---*---* - - - *---*
n columns

Suppose we cut the bar into $\displaystyle m$ horizontal strips with $\displaystyle m-1$ cuts. .** Code:

*---*---* - - - *---*
| | | | |
*---*---* - - - *---*
*---*---* - - - *---*
| | | | |
*---*---* - - - *---*
. .
. .
. .
*---*---* - - - *---*
| | | | |
*---*---* - - - *---*

Then each of the $\displaystyle m$ strips requires $\displaystyle n-1$ cuts to make unit squares.

. . This requires: .$\displaystyle m(n-1)$ cuts.

There will be a minimum of: .$\displaystyle (m - 1) + m(n-1) \;=\;\boxed{{\color{red}mn-1}\text{ cuts}}$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

If the initial cuts are made *vertically*, the numbers of cuts is the same.